X-ray imaging has been used in the medical field and for radiology in general, such as non-destructive testing and x-ray computed tomography. Conventional radiography systems use x-ray absorption to distinguish differences between different materials, such as normal and abnormal human tissues.
Conventional x-ray radiography measures the projected x-ray attenuation, or absorption, of an object. Attenuation differences within the object provide contrast of embedded features that can be displayed as an image. For example, cancerous tissues generally appear in conventional radiography because these tissues are more dense than the surrounding non-cancerous tissues. The best absorption contrast is generally obtained at x-ray energies where the absorption is high. Conventional radiography is typically performed using lower x-ray energy in higher doses to allow greater absorption and, thus, better contrast and images. Using x-rays having higher energy generally requires a lower dosage to be used because of patient safety concerns. In general, as the x-ray energy level increases and the x-ray dose used decreases, the quality of the conventional radiography image lessens.
Diffraction-enhanced imaging (DEI) and multiple-image radiography (MIR) are related phase-sensitive x-ray imaging methods, which generally use a system of diffracting crystals to analyze the angular components of an x-ray beam after it traverses an object. DEI can produce images depicting the effects of absorption and refraction of the beam by the object. MIR produces one additional image which shows the effect of ultra-small-angle scattering. A further advantage of MIR is that it uses a generally more-accurate imaging model.
The quantity depicted at each pixel in a DEI or MIR refraction-angle image is the angle Δθ by which an x-ray beam is refracted upon passing through the object. In the x-ray regime, the refractive index is always very nearly one; therefore, the measured refraction angles are very small. For example, refraction angles observed when imaging the breast are typically from about 0 to 1 μradians.
Thus, the x-component of the refraction angle can be represented by the following well-known small-angle approximation:
                                          Δθ            ⁡                          (                              x                ,                y                            )                                ≅                                    ∂                              ∂                x                                      ⁢                                          ∫                L                                                                              ⁢                                                n                  ⁡                                      (                                          x                      ,                      y                      ,                      z                                        )                                                  ⁢                                  ⅆ                  z                                                                    ,                            (        1        )            where L is the path traversed by the beam (which is assumed to be approximately a straight line), (x,y) are spatial coordinates describing the image domain, and z is the spatial coordinate along the beam propagation direction.
Equation (1) can also be written approximately in terms of mass density as follows:
                                                                        Δθ                ⁡                                  (                                      x                    ,                    y                                    )                                            ≅                            ⁢                              K                ⁢                                  ∂                                      ∂                    x                                                  ⁢                                                      ∫                    L                                                                                                  ⁢                                                            ρ                      ⁡                                              (                                                  x                          ,                          y                          ,                          z                                                )                                                              ⁢                                          ⅆ                      z                                                                                                                                              ≅                            ⁢                              K                ⁢                                                      ∂                                                                                                                      ∂                    x                                                  ⁢                                                      ρ                    T                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                                                                                          ≅                                ⁢                                  1.35                  ×                                      10                                          -                      6                                                        ⁢                                      λ                    2                                    ⁢                                                            ρ                      T                                        ⁡                                          (                                              x                        ,                        y                                            )                                                                                  ,                                                          (        2        )            where K=reλ2/4πu, re is the classical electron radius (2.82×10−5 Å), λ is the x-ray wavelength (in Å), u is the unified atomic mass unit (1.66×10−24 g), and ρT(x,y)∫Lρ(x,y,z)dz is the projected mass density of the object along path L (in g/cm3). Thus, since the x-ray wavelength λ is on the order of 1 Å, it is readily seen that the refraction angle Δθ is on the order of 1 μradian.
The refraction-angle image produced by DEI or MIR can be very detailed and informative. An example refraction-angle image of a breast tumor (invasive carcinoma) is shown in FIG. 1. In this image, each pixel's value is equal to the angle of refraction experienced by the portion of the beam incident at a given spatial location. Thus, in FIG. 1, bright values indicate regions where the beam is refracted to the left, and dark values indicate regions where the beam is refracted to the right.
As seen in FIG. 1, refraction-angle images generally exhibit high levels of detail in the object. In part, this is due to the derivative operator inherent in the physics (see Eqs. (1) and (2)), which produces an effect equivalent to computerized edge enhancement, but without the same sensitivity to noise. However, this advantage of the refraction-angle image is also a limitation. Whereas planar medical images traditionally measure the projection of some object property (such as absorption coefficient in radiography), the refraction-angle image represents the gradient of the projected mass density, which confounds information about the mass density and its spatial distribution. In addition, because the gradient removes the DC value of the signal, it discards absolute quantitative information.
To produce an image that is more quantitatively useful than the refraction-angle image, U.S. Pat. No. 7,076,025, issued to Hasnah et al., and herein incorporated by reference in its entirety, provides a mass density image from the refraction image. For example, Equation (2) can be inverted numerically to compute the projected mass density ρT(x,y), which has units of g/cm2. The same principle can be applied to obtain the projected refractive index nT(x,y)=∫Ln(x,y,z)dz.
The analytical solution of Equation (2) is simply:
                                                        ρ              T                        ⁡                          (                              x                ,                y                            )                                =                                                    1                K                            ⁢                                                ∫                  a                  x                                ⁢                                                      Δθ                    ⁡                                          (                                                                        x                          ′                                                ,                        y                                            )                                                        ⁢                                      ⅆ                                          x                      ′                                                                                            +            C                          ,                            (        3        )            where the integrating constant C=ρT(a,y) is a boundary condition representing the value of the projected density at the left edge of the image (i.e., at x=a). This boundary condition can be determined by imaging the object within a known medium, such as air, which provides a known reference at x=a.
Unfortunately, when even a modest level of noise is present, a numerical implementation of Equation (3) can yield significant artifacts (see examples in FIG. 2, center column). This limitation can be particularly relevant in clinical implementations of this technology, which may be photon-limited owing to the difficulty of producing small, bright x-ray sources. Therefore, there is a need to mitigate the effect of noise.